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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Finite element formulation of the general magnetostatic problem in the space of solenoidal vector functions
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by Mark J. Friedman PDF
Math. Comp. 43 (1984), 415-431 Request permission

Abstract:

A new finite element method for the solution of the general magnetostatic problem is formulated and analyzed. The space of trial functions consists of solenoidal piecewise polynomial vector functions. We start with an integral formulation, in terms of the flux density, in the domain occupied by magnetic material. Using the properties [10] of the spectrum of the relevant singular integral operator we derive a weak formulation involving an integral operator on the boundary only. Thus the resulting finite element matrix consists of a sparse part corresponding to the interior of the iron domain and a full part corresponding to the boundary. Using the method of monotone operators, existence and uniqueness of the solution of the weak formulation as well as its discretization are proven. Error estimates are derived with the special emphasis on the case when magnetic permeability is large. Finally, solution of the problem by successive iteration is analyzed.
References
    P. G. Akishin, S. B. Vorozhtsoz & E. P. Zhidkov, Calculation of the Magnetic Field of the Isochronous Cyclotron Sector Magnet by the Integral Equation Method, Proc. Compumag Conf., Grenoble, 1978. A. G. Armstrong, A. M. Collie, C. J. Diserens, N. J. Newman, M. Simkin & C. W. Trowbridge, New Developments in the Magnet Design Program GFUN, Rutherford Laboratory Report No. RL-5060. J. H. Bramble & J. E. Pasciak, "A new computational approach for the linearized scalar potential formulation of the magnetostatic field problem," IEEE Trans. Mag., v. Mag-18, 1982, pp. 357-361.
  • M. V. K. Chari and P. P. Silvester (eds.), Finite elements in electrical and magnetic field problems, Wiley Series in Numerical Methods in Engineering, John Wiley & Sons, Ltd., Chichester, 1980. MR 589746
  • Mark J. Friedman, Mathematical study of the nonlinear singular integral magnetic field equation. I, SIAM J. Appl. Math. 39 (1980), no. 1, 14–20. MR 585825, DOI 10.1137/0139003
  • Mark J. Friedman, Mathematical study of the nonlinear singular integral magnetic field equation. II, SIAM J. Numer. Anal. 18 (1981), no. 4, 644–653. MR 622700, DOI 10.1137/0718042
  • Mark J. Friedman, Mathematical study of the nonlinear singular integral magnetic field equation. II, SIAM J. Numer. Anal. 18 (1981), no. 4, 644–653. MR 622700, DOI 10.1137/0718042
    • M. J. Friedman & J. S. Colonias, "On the coupled differential-integral equations for the solution of the general magnetostatic problem," IEEE Trans. Mag., v. Mag-18, No. 2, March 1982, pp. 336-339. M. J. Friedman, Finite Element Formulation of the General Magnetostatic Problem in the Space of Solenoidal Vector Functions, Ph.D. Thesis, Cornell University, 1982.
  • Mark J. Friedman and Joseph E. Pasciak, Spectral properties for the magnetization integral operator, Math. Comp. 43 (1984), no. 168, 447–453. MR 758193, DOI 10.1090/S0025-5718-1984-0758193-1
    • R. Glowinski & A. Marrocco, "Numerical solution of two-dimensional magnetostatic problems by augmented Lagrangian methods," Comput. Methods Appl. Mech. Engrg., v. 12, 1977, pp. 33-46.
  • F. Hecht, Construction d’une base de fonctions $P_{1}$ non conforme à divergence nulle dans $\textbf {R}^{3}$, RAIRO Anal. Numér. 15 (1981), no. 2, 119–150 (French, with English summary). MR 618819
    • J. L. Lions & E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, 1972. M. H. Lean & A. Wexler, Accurate Field Computation with the Boundary Element Method, Proc. Compumag Conf., Chicago, 1982. B. H. McDonald & W. Wexler, "Mutually constrained partial differential and integral equation field formulation," in Finite Elements in Electrical and Magnetic Field Problems (M. V. U. Chari and P. Silvester, eds.), Wiley, New York, 1978.
  • I. D. Maegroĭz, Iteratsionnye metody rascheta staticheskikh poleĭ v neodnorodnykh, anizotropnykh i nelineĭnykh sredakh, “Naukova Dumka”, Kiev, 1979 (Russian). MR 533743
    • I. D. Mayergoyz, "On numerical investigation of magnetostatic field in nonlinear ferromagnetic media," Sbornik Kibernetika, Vychislitel’naya Tekhnika, v. 17, 1972. (Russian) J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967. J. C. Nedelec, Mixed Finite Elements in ${R^3}$, Rapport Interne no. 49, Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau, 1979.
  • Joseph E. Pasciak, An iterative algorithm for the volume integral method for magnetostatics problems, Comput. Math. Appl. 8 (1982), no. 4, 283–290. MR 679401, DOI 10.1016/0898-1221(82)90010-4
  • Joseph E. Pasciak, A new scalar potential formulation of the magnetostatic field problem, Math. Comp. 43 (1984), no. 168, 433–445. MR 758192, DOI 10.1090/S0025-5718-1984-0758192-X
    • I. I. Pekker, "Calculation of magnetic systems by integration over field sources," Izv. Vyssh. Uchebn. Zaved. Electromekh., v. 9, 1964, pp. 1047-1051 (Russian)
  • Walter Rudin, Principles of mathematical analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR 0385023
    • J. Simkin & C. W. Trowbridge, "On the use of total scalar potential in the numerical solution of field problems in electromagnetics," Internat. J. Numer. Methods Engrg., v. 14, 1979, pp. 423-440. J. Simkin & C. W. Trowbridge, Three-Dimensional Computer Program (TOSCA) for Nonlinear Electromagnetic Fields, Rutherford Laboratory Report No. RL-79-097. R. Temam, On the Theory and Numerical Analysis of the Navier Stokes Equations, North-Holland, Amsterdam, 1977. M. M. Vaĭnberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Halsted Press, New York-Toronto, 1973.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Math. Comp. 43 (1984), 415-431
  • MSC: Primary 78A30; Secondary 65N30, 78-08
  • DOI: https://doi.org/10.1090/S0025-5718-1984-0758191-8
  • MathSciNet review: 758191